A103633 → basis A013609 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-4 - 30\,x - 56\,x^{2} - 16\,x^{3} + 16\,x^{4} - 11\,n - 16\,n x + 24\,n x^{2} + 32\,n x^{3} + 9\,n^{2} + 24\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-40 - 212\,x - 388\,x^{2} - 240\,x^{3} + 64\,x^{4} + 96\,x^{5} - 26\,n - 8\,n x + 240\,n x^{2} + 416\,n x^{3} + 144\,n x^{4} - 64\,n x^{5} + 75\,n^{2} + 250\,n^{2} x + 200\,n^{2} x^{2} - 96\,n^{2} x^{3} - 128\,n^{2} x^{4} - 27\,n^{3} - 108\,n^{3} x - 144\,n^{3} x^{2} - 64\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 - 52\,x - 116\,x^{2} - 72\,x^{3} + 96\,x^{4} + 160\,x^{5} + 64\,x^{6} - 6\,n - 2\,n x + 116\,n x^{2} + 336\,n x^{3} + 352\,n x^{4} + 128\,n x^{5} + 18\,n^{2} + 102\,n^{2} x + 212\,n^{2} x^{2} + 192\,n^{2} x^{3} + 64\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	8	14	4						26
3	12	42	40	8					102
4	28	128	196	112	16				480
5	36	232	556	600	272	32			1728
6	80	600	1776	2616	1968	672	64		7776
7	96	904	3504	7176	8256	5184	1536	128	26784

All 15 rows

Technical details

Node id: n-57c9616c9698e557 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-7c9cde47112aa257

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   1;
  1:   3,    2;
  2:   8,   14,     4;
  3:  12,   42,    40,     8;
  4:  28,  128,   196,   112,    16;
  5:  36,  232,   556,   600,   272,     32;
  6:  80,  600,  1776,  2616,  1968,    672,    64;
  7:  96,  904,  3504,  7176,  8256,   5184,  1536,   128;
  8: 208, 2176,  9632, 23424, 33936,  29568, 14720,  3584,  256;
  9: 240, 2976, 15968, 48448, 91056, 108960, 81792, 36096, 7936, 512;

%F Formula

The row polynomials satisfy (9*n^2-11*n-4+(24*n^2-16*n-30)*x+(16*n^2+24*n-56)*x^2+(32*n-16)*x^3+16*x^4)*P(n,x) = -(27*n^3-75*n^2+26*n+40+(108*n^3-250*n^2+8*n+212)*x+(144*n^3-200*n^2-240*n+388)*x^2+(64*n^3+96*n^2-416*n+240)*x^3+(128*n^2-144*n-64)*x^4+(64*n-96)*x^5)*P(n-1,x) + (18*n^2-6*n-8+(102*n^2-2*n-52)*x+(212*n^2+116*n-116)*x^2+(192*n^2+336*n-72)*x^3+(64*n^2+352*n+96)*x^4+(128*n+160)*x^5+64*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 75*n^2 + 26*n + 40 + (108*n^3 - 250*n^2 + 8*n + 212)*x + (144*n^3 - 200*n^2 - 240*n + 388)*x^2 + (64*n^3 + 96*n^2 - 416*n + 240)*x^3 + (128*n^2 - 144*n - 64)*x^4 + (64*n - 96)*x^5)*P[n - 1, x] + (18*n^2 - 6*n - 8 + (102*n^2 - 2*n - 52)*x + (212*n^2 + 116*n - 116)*x^2 + (192*n^2 + 336*n - 72)*x^3 + (64*n^2 + 352*n + 96)*x^4 + (128*n + 160)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(9*n^2 - 11*n - 4 + (24*n^2 - 16*n - 30)*x + (16*n^2 + 24*n - 56)*x^2 + (32*n - 16)*x^3 + 16*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(27*n^3 - 75*n^2 + 26*n + 40 + (108*n^3 - 250*n^2 + 8*n + 212)*x + (144*n^3 - 200*n^2 - 240*n + 388)*x^2 + (64*n^3 + 96*n^2 - 416*n + 240)*x^3 + (128*n^2 - 144*n - 64)*x^4 + (64*n - 96)*x^5)*v[i - 1] + (18*n^2 - 6*n - 8 + (102*n^2 - 2*n - 52)*x + (212*n^2 + 116*n - 116)*x^2 + (192*n^2 + 336*n - 72)*x^3 + (64*n^2 + 352*n + 96)*x^4 + (128*n + 160)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(9*n^2 - 11*n - 4 + (24*n^2 - 16*n - 30)*x + (16*n^2 + 24*n - 56)*x^2 + (32*n - 16)*x^3 + 16*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

A064861 → D+1

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.